Finally, if \(\delta =\frac{1}{2}\)  and \(\rho =0\) , then by suitable choices of sequences \(N(d),k(d)\) , any limit in \([0,1]\)  can be achieved for \({\mathbb {E}}\,f_k(C_N)/\binom{N}{k}\) , and (hence) in general no limit exists. This can be deduced from the fact, following from the proof of part (a) of Theorem REF  and also shown in [1]}, that \(N-2d\sim c\sqrt{d}\)  and \(k\sim b\sqrt{d}\)  together imply
\( \lim _{d\rightarrow \infty } \frac{{\mathbb {E}}\,f_k(C_N)}{\binom{N}{k}} =\frac{\Phi \left(-(c+b)/\sqrt{2}\right)}{\Phi \left(-c/\sqrt{2}\right)}.\) 
